The first choice \(\rho _0\)  is used to evaluate the size of our small-uniform statistic
\(W_n\) , while \(\rho _1\)  and \(\rho _2\)  are used to evaluate power. The
second choice is a case where the coefficient vector is exactly spanned by the
first three eigenvectors of the Brownian motion covariance operator. The third
choice is an example where the coefficient vector cannot be linearly spanned by
those eigenvectors. We note [1]} also numerically
illustrates cases 1 and 3, while [2]} illustrates case
2. We fix the noise \(\varepsilon \)  distribution as a Gaussian \(\mathcal {N}(0,\sigma _\varepsilon ^2)\)  distribution where we pick variance \(\sigma _\varepsilon ^2 = \frac{1- \mathrm {snr}}{\mathrm {snr}} \mathrm {Var}(\langle X,\rho  \rangle )\) , and where \(\mathrm {snr}\) 
is the “signal-to-noise ratio” and we let it to be \(\mathrm {snr} = 5\%\)  and
\(10\%\) . To focus the discussion on the properties of our small-uniform
statistic and not on the estimation performance of an error estimator
\(\hat{\sigma }_\varepsilon \) , we assume throughout all these numerical
simulations that the noise parameter \(\sigma _\varepsilon ^2\)  is known with certainty.
